Q. 37 - Calculus

Question

Consider the region between the graph of $f\left(x\right)=x-2$ and the x-axis on [2,5]. For each line of rotation given in Exercises 35– 40, use definite integrals to find the volume of the resulting solid.  

Step-by-Step Solution

Answer

The required volume is $18\mathrm{\pi }$

Step 1. Given Information

The given figure is     

Step 2. Calculation

The external radius of each washer is 3 and internal radius of each washer is given as $3-f\left(x\right)$

Determine the volume of solid of revolution. 

$$\begin{gathered} V=\mathrm{\pi }{\int }_2^5\left(3^2-(3-\mathrm{f}{\left(\mathrm{x}\right))}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(3^2-(3-{(\mathrm{x}-2))}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(9-{(3-\mathrm{x}+2)}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(9-{(5-\mathrm{x})}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\int }_2^5\left(9-25+10\mathrm{x}-{\mathrm{x}}^2\right)\mathrm{dx} \\ =\mathrm{\pi }{\left[-16\mathrm{x}+10\frac{{\mathrm{x}}^2}{2}-\frac{{\mathrm{x}}^3}{3}\right]}_2^5 \\ =\mathrm{\pi }\left(\left(-16\left(5\right)[phone]\mathrm{}\frac{{\mathrm{}}^{\mathrm{}}}{\mathrm{}}\frac{{\mathrm{}}^{\mathrm{}}}{\mathrm{}}\right)\left(\mathrm{}\left(2\right)[phone]\mathrm{}\frac{{\mathrm{}}^{\mathrm{}}}{\mathrm{}}\frac{{\mathrm{}}^{\mathrm{}}}{\mathrm{}}\right)\right) \\ =18\mathrm{\pi } \end{gathered}$$